
diff -r 5f764272183e -r b2af2549efd1 src/HOL/IOA/IOA.thy
--- a/src/HOL/IOA/IOA.thy	Tue Jun 06 20:47:12 2006 +0200
+++ b/src/HOL/IOA/IOA.thy	Wed Jun 07 00:57:14 2006 +0200
@@ -195,6 +195,167 @@
         in
         ? x. Some(x) = ren(a) & (s,x,t):trans_of(ioa)})"
 
-ML {* use_legacy_bindings (the_context ()) *}
+
+declare Let_def [simp]
+
+lemmas ioa_projections = asig_of_def starts_of_def trans_of_def
+  and exec_rws = executions_def is_execution_fragment_def
+
+lemma ioa_triple_proj:
+    "asig_of(x,y,z) = x & starts_of(x,y,z) = y & trans_of(x,y,z) = z"
+  apply (simp add: ioa_projections)
+  done
+
+lemma trans_in_actions:
+  "[| IOA(A); (s1,a,s2):trans_of(A) |] ==> a:actions(asig_of(A))"
+  apply (unfold ioa_def state_trans_def actions_def is_asig_def)
+  apply (erule conjE)+
+  apply (erule allE, erule impE, assumption)
+  apply simp
+  done
+
+
+lemma filter_oseq_idemp: "filter_oseq p (filter_oseq p s) = filter_oseq p s"
+  apply (simp add: filter_oseq_def)
+  apply (rule ext)
+  apply (case_tac "s i")
+  apply simp_all
+  done
+
+lemma mk_trace_thm:
+"(mk_trace A s n = None) =
+   (s(n)=None | (? a. s(n)=Some(a) & a ~: externals(asig_of(A))))
+   &
+   (mk_trace A s n = Some(a)) =
+    (s(n)=Some(a) & a : externals(asig_of(A)))"
+  apply (unfold mk_trace_def filter_oseq_def)
+  apply (case_tac "s n")
+  apply auto
+  done
+
+lemma reachable_0: "s:starts_of(A) ==> reachable A s"
+  apply (unfold reachable_def)
+  apply (rule_tac x = "(%i. None, %i. s)" in bexI)
+  apply simp
+  apply (simp add: exec_rws)
+  done
+
+lemma reachable_n:
+  "!!A. [| reachable A s; (s,a,t) : trans_of(A) |] ==> reachable A t"
+  apply (unfold reachable_def exec_rws)
+  apply (simp del: bex_simps)
+  apply (simp (no_asm_simp) only: split_tupled_all)
+  apply safe
+  apply (rename_tac ex1 ex2 n)
+  apply (rule_tac x = "(%i. if i<n then ex1 i else (if i=n then Some a else None) , %i. if i<Suc n then ex2 i else t)" in bexI)
+   apply (rule_tac x = "Suc n" in exI)
+   apply (simp (no_asm))
+  apply simp
+  apply (rule allI)
+  apply (cut_tac m = "na" and n = "n" in less_linear)
+  apply auto
+  done
+
+
+lemma invariantI:
+  assumes p1: "!!s. s:starts_of(A) ==> P(s)"
+    and p2: "!!s t a. [|reachable A s; P(s)|] ==> (s,a,t): trans_of(A) --> P(t)"
+  shows "invariant A P"
+  apply (unfold invariant_def reachable_def Let_def exec_rws)
+  apply safe
+  apply (rename_tac ex1 ex2 n)
+  apply (rule_tac Q = "reachable A (ex2 n) " in conjunct1)
+  apply simp
+  apply (induct_tac n)
+   apply (fast intro: p1 reachable_0)
+  apply (erule_tac x = na in allE)
+  apply (case_tac "ex1 na", simp_all)
+  apply safe
+   apply (erule p2 [THEN mp])
+    apply (fast dest: reachable_n)+
+  done
+
+lemma invariantI1:
+ "[| !!s. s : starts_of(A) ==> P(s);
+     !!s t a. reachable A s ==> P(s) --> (s,a,t):trans_of(A) --> P(t)
+  |] ==> invariant A P"
+  apply (blast intro!: invariantI)
+  done
+
+lemma invariantE:
+  "[| invariant A P; reachable A s |] ==> P(s)"
+  apply (unfold invariant_def)
+  apply blast
+  done
+
+lemma actions_asig_comp:
+  "actions(asig_comp a b) = actions(a) Un actions(b)"
+  apply (auto simp add: actions_def asig_comp_def asig_projections)
+  done
+
+lemma starts_of_par:
+  "starts_of(A || B) = {p. fst(p):starts_of(A) & snd(p):starts_of(B)}"
+  apply (simp add: par_def ioa_projections)
+  done
+
+(* Every state in an execution is reachable *)
+lemma states_of_exec_reachable:
+  "ex:executions(A) ==> !n. reachable A (snd ex n)"
+  apply (unfold reachable_def)
+  apply fast
+  done
+
+
+lemma trans_of_par4:
+"(s,a,t) : trans_of(A || B || C || D) =
+  ((a:actions(asig_of(A)) | a:actions(asig_of(B)) | a:actions(asig_of(C)) |
+    a:actions(asig_of(D))) &
+   (if a:actions(asig_of(A)) then (fst(s),a,fst(t)):trans_of(A)
+    else fst t=fst s) &
+   (if a:actions(asig_of(B)) then (fst(snd(s)),a,fst(snd(t))):trans_of(B)
+    else fst(snd(t))=fst(snd(s))) &
+   (if a:actions(asig_of(C)) then
+      (fst(snd(snd(s))),a,fst(snd(snd(t)))):trans_of(C)
+    else fst(snd(snd(t)))=fst(snd(snd(s)))) &
+   (if a:actions(asig_of(D)) then
+      (snd(snd(snd(s))),a,snd(snd(snd(t)))):trans_of(D)
+    else snd(snd(snd(t)))=snd(snd(snd(s)))))"
+  (*SLOW*)
+  apply (simp (no_asm) add: par_def actions_asig_comp Pair_fst_snd_eq ioa_projections)
+  done
+
+lemma cancel_restrict: "starts_of(restrict ioa acts) = starts_of(ioa) &
+              trans_of(restrict ioa acts) = trans_of(ioa) &
+              reachable (restrict ioa acts) s = reachable ioa s"
+  apply (simp add: is_execution_fragment_def executions_def
+    reachable_def restrict_def ioa_projections)
+  done
+
+lemma asig_of_par: "asig_of(A || B) = asig_comp (asig_of A) (asig_of B)"
+  apply (simp add: par_def ioa_projections)
+  done
+
+
+lemma externals_of_par: "externals(asig_of(A1||A2)) =
+   (externals(asig_of(A1)) Un externals(asig_of(A2)))"
+  apply (simp add: externals_def asig_of_par asig_comp_def
+    asig_inputs_def asig_outputs_def Un_def set_diff_def)
+  apply blast
+  done
+
+lemma ext1_is_not_int2:
+  "[| compat_ioas A1 A2; a:externals(asig_of(A1))|] ==> a~:internals(asig_of(A2))"
+  apply (unfold externals_def actions_def compat_ioas_def compat_asigs_def)
+  apply auto
+  done
+
+lemma ext2_is_not_int1:
+ "[| compat_ioas A2 A1 ; a:externals(asig_of(A1))|] ==> a~:internals(asig_of(A2))"
+  apply (unfold externals_def actions_def compat_ioas_def compat_asigs_def)
+  apply auto
+  done
+
+lemmas ext1_ext2_is_not_act2 = ext1_is_not_int2 [THEN int_and_ext_is_act]
+  and ext1_ext2_is_not_act1 = ext2_is_not_int1 [THEN int_and_ext_is_act]
 
 end




